Analytical modification of structural natural frequencies

FINITE ELEMENTS IN ANALYSIS A N D DESIGN

ELSEVIER

Finite Elements in Analysis and Design 18 (1994) 75 81

Analytical modification of structural natural frequencies Stanley G. H u t t o n Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC V6T 1Z4 Canada

Abstract

This paper concerns the analysis of the frequency characteristics of vibrating structural systems and the procedures that are available for modification of these systems. In such problems it is common to modify a baseline model in order to change its modal characteristics in some desired manner. Analytical procedures that do not rely upon a complete reanalysis of the baseline system are discussed, including inverse modification approaches that produce the physical changes required to effect a prescribed frequency modification as output.

1. Introduction

In the development of many structures, machines, or components thereof, an important consideration for a successful design is that the vibration response of the system should be within specified limits. At the analysis stage it is common to construct a finite element model, compute the forced response, investigate the modal characteristics of the system, or, more simply, compare natural frequencies with known forcing frequencies. In the event that a vibration problem exists, the question arises as to how the characteristics of the structure may be modified in such a manner as to minimize the problem. The present paper is concerned with a discussion of the various analytical procedures that are available for dealing with such problems. Attention will be confined to systems whose forced response can be represented by an equation of the form

M Yc(t) + Kx(t) = F(t),

(1)

where M, K, and x(t) represent the mass matrix, stiffness matrix and vector of generalized coordinates, and F(t) is a vector of applied forces. It is assumed that M is symmetric positive definite and that K is symmetric positive definite or semi-definite. Such a system has natural frequencies o21, m2, ..., 09N and corresponding mode shapes T1, T2 . . . . . q~N; it will be assumed that all the natural frequencies are distinct. The correspondence between the frequency content of F(t) and the values of the natural frequencies has a primary effect upon the response of the system. If, for 0168-874X/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 8 - 8 7 4 X ( 9 4 ) 0 0 0 1 6 - 9

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example, close correspondence exists between a forcing frequency and one of the natural frequencies, the designer will be interested in changing the forcing frequency or modifying the natural fequencies of the baseline system. If it is desired to change the frequency characteristics of the structure by making mass and/or stiffness modifications, the resulting equations may be expressed in the form (M + AM)2(t) + (K + AK)x(t) = F(t),

(2)

where AM and AK represent the effect of the changes on the global mass and stiffness matrices. The frequencies coi and mode shapes qJl corresponding to the modified structures are defined by [(K + AK) - co'i2(M + AM)]qJ'~ = 0,

(3)

and a central question is that of determining AK and AM such that the resulting frequencies ~oi (and mode shapes ~ i ) correspond to prescribed values chosen to minimize the vibration problem. In the present paper, attention will be restricted to the case in which it is required to modify a natural frequency of the structure. In 1985 Baldwin and Hutton [1] presented a review of procedures available for modification of vibration characteristics and this present paper summarizes further work conducted in this area by the author.

2. Modification approaches Two basic approaches are available. In the first, changes are made to the structure based upon physical intuition and the resulting equations are solved to check if the prescribed changes had the desired effect. If not, further changes are made and the process is repeated. Such procedures may be termed forward modification procedures. This trial and error approach can be made relatively efficient by the use of appropriate modal truncation procedures as will be described. In the second approach, algorithms are developed in which the prescribed changes to the natural frequency are specified as input data and the output defines the mass and stiffness changes required to effect the required modification. Such procedures are called inverse modification procedures. 2.1. Forward modification using modal truncation

Consider the baseline system defined by Eq. (1). The solution of the free vibration problem provides the lowest n natural frequencies and mode shapes where, in general, n may be very much less than N, the total number of degrees of freedom of the problem. Let kIJ = [kI'/1, kI'/2 .....

(4)

~l'/n].

Assuming that the response of the modified system may be expressed as (5)

x(t) = U/ p(t),

substituting into Eq. (2) and premultiply by ~

leads to

(I + udx AM~F))5(t) + (f~ + UdXAKUd)p(t) = ~TF(t),

(6)

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where ~ is a diagonal matrix containing the lowest n natural frequencies squared. The mode shapes are assumed to be normalized with respect to the baseline mass matrix. Eq. (6) is a set of n equations with symmetric coefficient matrices that constitutes an approximation to the N equations of the complete modified problem, as expressed in Eq. (2). In many cases it is only-low frequency response that is interest, and thus n <
2.2. Inverse modification procedures

With increased use of finite element analysis, perturbation methods have become popular for inverse modification problems. The philosophy of peturbation methods is to investigate the solution of a modified structure by considering the modification as a perturbation of the linear baseline system. Stetson [2-4] used a linearized perturbation equation in which all nonlinear incremental terms were discarded. In addition, the changes in the structural mode shapes were described by a linear combination of the baseline modes. This allows first-order approximate solutions to be found for frequency and mode shape modification problems. Various refinements of this approach [5-9] have been made since the original work of Stetson. Hoff et al. [6] presented a two-stage predictor-connector method for frequency modification. In this method an additional perturbation analysis, based on the linear analysis, is performed to improve the first-order estimate of the structural changes. It has been found by Welch [10] that the predictor-corrector scheme is adequate for problems with linear property changes but is unable to compute accurately nonlinear property changes. Kim et al. [1 1] proposed a method in which the nonlinear equations of the modified structure are solved using mathematical programming. More general inverse eigenvalue problems have been studied involving the specification of one or more eigenvalues of a matrix and the evaluation of how the coefficients of the matrix need to change to meet the prescribed eigenvalues [-12-16]. Friedland et al. [15] gives a review of various applications of inverse eigenvalue problems and describes four general method for solving them. All of these use a Newton method approach in which approximate solutions are computed iteratively until they converge to the correct result. Three different algorithms are used to update to the eigenvectors: (i) recalculating with an eigenvalue analysis; (ii) updating with inverse iteration; (iii) updating by evaluating the participation of the baseline vectors in the modified vector. This last method has been used in perturbation methods [3-7], but it has been found that large mode shape changes are not predicted accurately. This is because the participation factor are computed from a linearized equation, which is only accurate for small changes. In the inverse frequency modification problem it is required to determine what structural changes will produce a prescribed changed in a given frequency. This problem is complicated by the fact that, in general, there is no simple relationship between frequency change and structural change. The rate of change of the ith baseline eigenvalue 21 at the baseline configuration with

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respect to the rth design parameter p, is given by

~Pr -- m i

~Pr -- "~i

(7)

~i,

which constitutes an exact relationship for the rate of change of 2i at the baseline configuration. In general, K and M will be functions of m design variables; whence,

(8) This equation provides an approximation to the change of the ith eigenvalue due to changes Ap, in the design variables. It will be recognized that all the terms on the right-hand side of Eq. (8) are functions of the variables p, and the approximation approaches an exact relationship if Apr -~ 0. Thus, for finite values of Ap, this equation must be solved iteratively with appropriate stiffness and mass matrices, and mode shape updating at each stage of the iteration. Smith and Hutton [17] discuss the use of Newton's method and inverse iteration in this context. As an illustration of the use of this formulation, consider the structure shown in Fig. 1. Shown in this figure is a finite element model with nine element groups, with each group consisting of three axisymetric beam elements. At the base, the beams are restrained against displacement and rotation. Also shown is the third mode of vibration which has a frequency of 7.84 Hz. As an illustration, it is required to raise the third natural frequency by 10, 30, 50, 100 and 200%. Assume that the design parameter is the second moment of area of beam elements and three independent parameters are choosen: ~1

]

//~i t

;'~"

.

7

",, ,':~

2

8

-__._.~

',,'~

Fig. 1. Finite element model of a mast structure: (a) definition of element groups (b) third mode of vibration.

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representing the change to elements 1, 4, and 7; a2 representing the change to 2, 5, and 8; and ~3 the change to 3, 6, 9. Thus, there are three independent parameters and one frequency constraint to find. Table 1 presents the results obtained by using Eq. (8) as implemented in [17]. As may be noted, the prescribed modification to the third frequency is achieved. As expected, the larger the frequency shift the larger the number of iterations required. The largest design changes are made to the ~3 group. This is consistent with the physics of the problem as it is this element set that accounts for the largest proportion of strain energy in this mode. The inverse modification problem has also been framed by Smith and Hutton [18] as a perturbation problem from which the changes required in order to produce arbitrary sized frequency modifications can be predicted exactly for certain conditions. In this approach, Eq. (3) is multiplied by a baseline eigenvector Wi and the modified eigenvector is expressed as a linear combination of these baseline vectors as /1

(9)

WI = Y~ CikWk = WCi k=l

Such substitution leads to a system of equations of the form WV[AK-2'iAM]Wci= -(A-21I),

(10)

c i = 1.... ,n,

where A is a diagonal matrix of baseline eigenvalues. The stiffness and mass matrix changes AK and AM need to be related to the changes in the model. Such changes are most usefully expressed in terms of physical design variables. Such changes may lead to changes in the mass and stiffness matrices that are linearly or non-linearly related to the design variables. For example, changing the thickness of plate elements causes the stiffness matrix to change as a non-linear function of the thickness, whereas a change to the thickness of membrane elements results in a linear change to the stiffness matrix. If attention is confined to design variables upon which AK and AM depend linearly, and if m

m

AK = ~ Kjctj,

AM = Y, Mj~j,

j=l

j=l

(11)

Table 1 Results for frequency modification of mast structure Prescribed frequency ratio

no. iterations

Value of design variables

Modified frequency ratios

f~/f3

~1

°~a

o~3

Y',a~

f'*/fl

f'z/fz

f'3/f3

1.10 1.30 1.50 2.00 3.00

0.059 0.247 0.535 1.50 14.9

0.330 1.36 2.71 590 17.8

0.635 2.87 6.90 28.9 92.8

0.516 10.1 55.2 871 9140

1.04 1.14 1.26 1.57 2.04

1.04 1.14 1.26 1.57 2.04

1.10 1.30 1.50 2.00 3.00

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Table 2 Results for frequency modification of cantilevered beam Prescribed frequency ratio

1.1

1.5

2.0

~1 ~2 ~3 ~4 ~s ~6

0.289 0.175 0.086 0.030 0.005 0.000

1.727 1.173 0.682 0.289 0.064 0.003

4.175 2.918 1.864 0.904 0.224 0.010

Modified frequency ratio

1.100

1.496

1.997

where ~j defines the magnitude of the change of the jth variable and K j, M r define global stiffness and mass sensitives to ~j, then ?n

~teV[Kj-

2'iMj]te c, = - ( A - 2'~I)c~,

i = 1. . . . . n.

(12)

j=l

These equations contain complete information on the relationship between the design variable c~j, the frequencies of the modified structure 2) and the mode shapes of the modified structure tel. If a complete set of mode shapes te~ is available, the exact solutions can be found irrespective of the magnitude of the changes of the design variable. In general, only a subset of the modes will be available and approximate solutions will be obtained based upon a work balance between the elastic and inertial forces involved in the motion of the modes available. In order to modify the frequency of the kth mode, the k equation from (12) is chosen. Specifying that 2~, = 2* gives ~ ~ j te[ [ K j - )~* m j]

t e k ~- ( ~'~ - - "~k) Ckk "

(13)

j=l

By choosing the kth equation from (12), the algorithm seeks to change the frequency spectrum such that the kth frequency of the modified modes attains the prescribed value. As an example of the use of Eq. (13), the frequencies of a simple uniform cantilevered beam are modified. The baseline modes retained consist of the lowest three bending modes, i.e., n = 3. The objective is to raise the fundamental frequency by 10%, 50% and 200%. The beam is modeled with six elements and the design variables 0~1-~6 are the second moments of area of each element (0~6 representing the element at the free end). Table 2 shows the results and, as may be seen, the prescribed frequency modifications are achieved with high accuracy using changes that are physically reasonable (the root element has the largest modification).

3. Summary Analytical procedures for modifying the vibration characteristics of structures have been presented. It has been shown that once a baseline analysis has been conducted the calculated modes can be effectively used in further computation to determine how to modify the structure to obtain the

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prescribed frequencies. During forward modification modal condensation can be used to decrease the analysis time significantly and thus allow the designer to evaluate more modification alternatives in a given time. The baseline modal information can also be used to construct perturbation-based inverse modification formulations which can be used to determine accurately the mass and stiffness changes required to modify specified natural frequencies.

Acknowledgement Funding for much of this work was provided by the Canadian Department of Natural Defense, Defense Research Establishment Atlantic, Halifax, Novia Scotia.

References [1] J.F. Baldwin and S.G. Hutton, "Natural modes of modified structures", A I A A J. 23, pp. 1737 1743, 1985. [2] K.A. Stetson, "Perturbation method of structural design relevant to holographic vibration analysis", A I A A J. 13, pp. 457-459, 1975. [3] K.A. Stetson and G.E. Palma, "Inversion of first-order perturbation theory and its application to structural design", A I A A J. 14, pp. 454-460, 1976. [4] K.A. Stetson, I.R. Harrison and G.E. Palma, "Redesigning structural vibration modes by inverse perturbation subject to minimal change Theory", Comput. Method Appl. Mech. Eng., 16, pp. 151 175, 1978. [5] R.E. Sandstrom and W.J. Anderson, "Modal perturbation methods for marine structures", S N A M E Trans. 90, pp. 41-54, 1982. [6] C.J. Hoff, M.M. Bernitsas, R.E. Sandstrom and W.J. Anderson, "Inverse perturbation method for structural redesign with frequency and mode shape constraints", AIAA 22, pp. 1304 1309, 1984. [7] C.J. Hoff and M.M. Bernitsas, "Dynamic redesign of marine structures", J. Ship Res. 29, pp. 285-295, 1985. [8] M.M. Bernitsas and B. Kang, "Admissible large perturbations in structural redesign", A I A A J. 29, pp. 104-113, 1991. [9] H.D. Gans and W.J. Anderson, "Structural optimization incorporating centrifugal and Coriolis effects," AIAA J. 29, pp. 1743-1750, 1991. [10] P.A. Welch, "Dynamic redesign of modified structures", M.A.Sc. Thesis, University of British Columbia, Vancouver, Canada, 1987. [11] K. Kim, W.J. Anderson and R.E. Sandstrom, "Nonlinear inverse perturbation method in dynamic analysis", AIAA J. 21, pp. 1310-1316, 1983. [12] Biegler-Konig, and W. Friedrich "A Newton iteration process for inverse eigenvalue problems", Numer. Math. 37, pp. 349-354, 1981. [13] Z. Bohte, "Numerical solution of the inverse algebraic eigenvalue problem," Comput. J. 10, pp. 385-388, 1967-1968. [14] A.C. Downing and A.S. Householder, "Some inverse characteristic value problems", J. Assoc. Comput. Mach., 3, pp. 203-207, 1956. [15] S. Friedland and J. Nocedal and M.L. Overton, "The formulation and analysis of numerical methods for inverse eigenvalue problems", S l A M J. Numer. Anal. 24, pp. 634-667, 1987. [16] M.R. Osborne, "On the inverse eigenvalue problem for matrices and related problems for difference and differential equations", Conference on Application of Numerical Analysis, Lecture Notes in Math, No. 228, Springer, Berlin, New York, pp. 155-168, 1971. [17] M.J. Smith and S.G. Hutton, "Frequency modification using Newton's method and inverse iteration eigenvector updating", A I A A J. 30, pp. 1886 1891, 1992. [18] M.J. Smith and S.G. Hutton, "A perturbation method for inverse modification of discrete undamped systems", A S M E J. Appl. Mech., to appear.