IDENTIFICATION OF THE MODAL PROPERTIES OF AN ELASTIC STRUCTURE FROM MEASURED TRANSFER FUNCTION DATA

ABSTRACT

A technique for identifying the modal properties of an elastic structure in a testing laboratory is presented. The technique is based upon the use of digital processing and the fast Fourier transform (FFT) to obtain transfer function data, and then the use of a least squared error estimator to identify modal properties from the transfer function data. Both analytical and experimental results are presented.

INTRODUCTION

In recent years the implementation of the fast Fourier transform (FFT) in low cost mini­computer systems has provided the environmental testing laboratory with a faster and more powerful tool for acquisition and analysis of vibration data. from mechanical structures. The results are used by analysts and designers alike as an aid to better understanding and improving mechanical designs.

In this paper an analytical technique is presented which as been implemented in a Fourier Analyzer to provide modal data on site in a testing laboratory. The technique is based upon the application of a least squares estimator to measured transfer function data. During the process the natural frequencies, damping factors, and mode shapes of all the predominant modes of vibration of a structure are identified.

A brief review of the modal theory and a derivation of the analytical form used in the estimation process are given in the following section. Following that is a discussion of how parameters are obtained from a single transfer function, and some experimental results are given. Lastly the global nature of a mode is discussed and verified with experimental results.

MODAL THEORY

Assume that an elastic system has n­degrees of freedom and that its motion can be adequately described by n­linear differential equations with constant coefficients. written as

M d²x(t)/dt² + C dx/dt(t) + Kx(t) = ft) (1)

where x(t) and f(t) are displacement and force n­vectors respectively, and M, C, and K are real symmetric matrices. M is called the mass matrix, C the damping matrix, and K the stiffness matrix.

Taking the Laplace transform of equation (1) gives

where X(s) <­­> x(t) and F(s) <­­> f(t) are vector Laplace transform pairs. B is defined as the (n x n) system matrix. Eq. (2) is often referred to as an expression of the dynamic flexibility of the structure.

Eigenvectors (y_k) and eigenvalues () of the matrix can be defined in the usual way, i.e. to satisfy the equation

where y_k is an n­vector and is a constant. The system Matrix B has (n) eigenvalues and (n) eigenvectors; each eigenvalue­vector pair is defined by equation (3).

It is straightforward to show that the y_k eigenvectors are orthogonal, provided all values are different, as follows:

For two different eiqenvalues (k) and (j)

Note that equation (5) can be rewritten as

since B is symmetric. (The superscript t denotes the transpose operation.) Pre­multiplying equation (4) by y^t_j. and post multiplying equation (6) by y_k

so

Thus for not equal , y^t_j y_k = 0 which defines orthogonality between two eigenvectors.

As usual, the eigenvalues can be expressed as roots of the determinant

where I is an (n x n) identity matrix, and

each value can be found by solving the polynomial equation defined by (9).

We define a transformation matrix as

where the n columns of are the eigenvectors y_k. We also define a diagonal matrix of eigenvalues as

Then, the above definition of eigenvalues and eigenvectors can be expressed in matrix form as

By defining the generalized inverse of as

equation (12) can be rewritten as

If the eigenvectors are normalized to unit magnitude, so that = I, then =. In any case, is an (n x n) diagonal form of B. The general form D^t BD is called a congruence transformation, and if the columns of D are orthogonal (so that -D^tD is diagonal), it is called an orthogonal transformation. Thus,

B = represents an orthogonal diagonalization of B. If the eigenvalues of B are unique, then the eigenvectors are also unique except for an arbitrary normalization constant, so this orthogonal diagonalization of B must be unique to the same extent.

The transfer function matrix H of this linear system (1) is defined as

assuming that the indicated matrix inverses exist.

We can also write

Thus, is the orthogonal diagonalization of H. which is unique except for normalization constants. Note that both B and H are diagonalized by the same orthogonal transformation.

Recall that the elements of B are quadratic functions of s. Both the eigenvalues and the eigenvector components y_k are generally rather complicated (usually irrational) functions of s. This means that the eigenvector components in the time domain are each changing in some complicated way with respect to one another, and that each corresponding eigenfunction (time domain representation of ) is a complicated time waveform. The only real advantage to this formulation is that each eigenvector distribution is orthogonal with respect to all other eigenvectors.

It is preferable to decompose B or H into a set of time invariant vectors (independent of s), and put all s dependence into some diagonal representation of the system or transfer matrix. Practical experience indicates that this possibility exists, i.e. physical structures exhibit "standing wave" vibration patterns at certain frequencies, in which a "global" vibration mode shape is associated with each "resonant" frequency. We are further encouraged by the fact that the solution of the homogeneous wave equation can be expressed as the product of a time function and a space function. Finally the driving function can be decomposed into a linear combination of these homogeneous solutions, and the complete solution obtained in terms of linear combinations of these homogeneous "eigenfunctions". It should be apparent that the key to this desired decomposition of B lies in the solution to the homogeneous equation Bu = 0.

It is now shown how this homogeneous equation can be solved in terms of the previously defined eigenvalues and eigenvectors of B. We begin by recognizing that each element of H = B^-1 is a rational fraction in s, with denominator given by detail. Thus, the roots of this denominator, called the poles of H. are the values of s = s_k for which det|B| = 0. These values of s also satisfy the above homogeneous equation Bu = 0.

Each element of H can be expanded into a partial fraction expansion about each pole so that H can be written in the following form:

where the a_k's are matrices independent of s. Recall the representation of H in terms of eigenvalues and orthogonal eigenvectors.

Each a_k matrix can be found by multiplying H times s­s_k, and then setting s = s_k, provided all S_k values are different. Thus,

Recognize that there are 2n poles because each element of B is of quadratic form. Further, the poles generally appear in complex conjugate pairs because the elements of M, C, and K are real numbers, and hence each quadratic element of B has real coefficients. If poles do not appear in conjuqate pairs, then they must be real.

Now, is a diagonal matrix whose elements are functions of s. Furthermore,

because is similar to B ( = B), and hence has the same eigenvalues. Thus, any value of s = s_k which satisfies det [B] = 0, will also force one of the 's, say, to zero. Rewriting the eigenvector definition

Thus, = 0 for either s = s_k or s = s^*_k , and the homogeneous solution at s^k= s_k is the original eigenvector evaluated at s = s_k. Also, the solution corresponding to the conjunate poles

Note that u_k u^t_k is an (n x n) symmetric (complex) matrix, while u^t_k u_k is a complex scalar.

Therefore, the a_k matrix is determined by a mode shape vector u_k, which is simply the solution to the homogeneous system equation B u_k = 0 for s = s_k . Furthermore, each column of a_k is this same bode shape vector (within a constant multiplier), and each row is the transpose of the vector. From a measurement standpoint this implies that the same mode shape is obtained regardless of which spatial point is excited or monitored. This pervasiveness of the mode shapes throughout the transfer matrix is verified with experimental results later in the paper.

Returning to the partial fraction expansion of H.

We can represent this summation of partial fraction terms in matrix form by defining the following matrices:

is called the modal transformation matrix, and ^t is the transfer matrix in modal coordinates. Note that the columns of are not orthogonal (even though the parent eignevectors y_k are orthogonal) because each u_k is evaluated at a different value of s. However, the elements of are not functions of s. All of the s dependence is contained in. Each column of represents a normalized mode shape vector for the corresponding pole of H. It should be apparent that this normalization is arbitrary, and could be absorbed into the matrix if desired.

As discussed previously, the poles of H usually occur in conjugate pairs, and for this case the mode shape vectors associated with the negative poles (lower half of s­plane) are simply the conjugates of the vectors associated with the positive poles. Thus, if ₁^*, is defined as that (n x n) part of associated with positive poles, then of will correspond to the negative poles. Similarly, can be broken into two parts, ₁ comprising the positive poles, and A2 comprising the negative poles. H can then be represented

or in partitioned form as

Each of these submatrices is (n x n) and only ₁ and ₂ are functions of s.

Define

we define the modal mass as the coefficient of s² ien the denominator of each element of H. However, we recognize that this coefficient is arbitrary, depending on the numerator normalization. Notice that A_k has the dimensions of (s.mass)^-1, so the numerator should be normalized by dividing by something of the form A_kS_k. We can use the rather arbitrary normalization factor

Notice that each element of the H matrix has a different zero in the s-plane, depending upon the angle of A_k and u_k at each point, but the poles of each element of H are common, and occur at s = s_k and s = s_k*.

For the special case of zero damping (ck = 0),

called the normal mode case, we find that s_k =­ s_k* is purely imaginary. Thus, the B matrix becomes real symmetric, and it's eigenvalues and eigenvector components become real. This means that u_k = u_k^* , and we can show that A_k becomes purely imaginary, so A_k =­ A_k*. In this case, the numerator zero in each element of H goes to infinity, and H becomes

Thus it has been shown that two transfer function forms of interest in modal analysis, namely the complex eigenvalue­eigenvector case (eq. 34) and the normal mode case (eq. 35) can be obtained from a more general eigenvalue­eigenvector diagonalization of the system or transfer matrix. In the next section the identification of modal parameters from measured transfer function data using eq. 34 is discussed.

IDENTIFICATION OF MODAL PARAMETERS

The technique used to obtain the results presented here involves the curve­fitting of analytical expression (34) to a set of measured transfer function data. The curve fitting is performed in a manner which minimizes the squared difference between the complex data and the complex valued analytical function form, i.e. a least squared error estimate of the data is determined.

Recall that according to the modal theory, only one row or one column of the transfer matrix need be measured since all other rows and columns contain redundant information. During the process of determining the least squared estimator for the transfer matrix, complex values of S_k and the residues of one column or one row of the transfer matrix H for all predomenant modes of vibration are determined.

For example, the q^th column of H would

have the residues

After measuring these n residues of H, we form the sum of the squares of these numbers giving

Taking the square root, and normalizing the measured residues by this quantity gives

,

which are the elements of the normalized mode shape vector. The A_k coefficients are readily found from any residue of H_pq by dividing by the product of the p^th and q^th components of the normalized mode shape vector. The modal system parameters (mass, stiffness, damping) are obtained from A_k and s_k, and the mode shape is given by the u_k vectors (generally normalized by

.

The pole location of mode (k) in the s­plane, also called the complex frequency, can be written in terms of the coordinates

is called the damping factor and w_k the natural frequency of mode (k). Other related and commonly used terms are the damping ratio and resonant frequencv.

These terms are shown in the s­plane in figure (1)

The experimental data was taken from the metal T-plate mounted on a foam rubber base shown in Figure 2.

A hammer was used to provide the broad band excitation force, with a load cell attached to it to measure the force. An accelerometer mounted on the plate was used to measure responses.

The transfer function data was obtained using a Hewlett­Packard 5451B Fourier Analyzer, and the modal parameter identification ::as performed using the Hewlett­Packard Modal Analysis Package.

Transfer functions were measured between 22 different points evenly spaced along the outer periphery of the T­plate. Figure 3 shows a typical transfer function in rectangular or co­quad form.

Figure 4 shows the least squares estimate of this transfer function and Table 1 contains its corresponding modal parameters. These results were generated on the Fourier Analyzer using the Modal Package in about 30 seconds.

A MODE AS A GLOBAL PROPERTY

By far the most fundamental assumption of modal testing is that a mode of vibration can be excited from anywhere on an elastic structure, except of course along its node lines where it can't be excited al all. This is another way of stating the result derived earlier, i.e. that the same mode shape vector (scaled by a different component of itself) is contained in every row and column of the transfer matrix. In addition, modal frequency and damping are constants which can be identified in any element of the transfer matrix, i.e. any transfer function taken from the structure.

It is important to recognize that this global mode shape concept implies some sort of spatial boundaries, beyond which vibrations will not readily propagate. Any attempt to extent B or H beyond these boundaries will result in singular matrices, and a breakdown of the modal concept. This behavior implies that B and H must be partitioned into sub­matrices, some of which will be nonsingular, and will possess well defined vibration modes. If two linear systems are completely isolated, then a single composite mode including both systems is not meaningful.

Conversely, it is important to include enough spatial points to describe all of the vibration modes of interest. If some region of a bounded system is not monitored or excited, or if points are not chosen sufficiently close together, then some modes cannot be adequately represented.

Following are the results of two separate modal tests that were performed on the T­plate. In test #1 the accelerometer was mounted on the bottom plate as shown in figure 2. and the plate was impacted with the hammer at the 22 peripheral locations. Using the Fourier Analyzer a transfer function was measured between each of the 22 impact points and the single response point (accelerometer location). Since the transfer function is the same between two points regardless of which one is the excitation or response point this test is equivalent to impacting the plate in one spot and moving the accelerometer to all 22 positions. This reciprocity or symmetry assumption is also fundamental to modal analysis and is reflected in the symmetry of the system and transfer matrices.

Test #2 was the same as test #1 except that the accelerometer was mounted at position #2.

Table 2. contains the least squared estimates of the natural frequency and damping factor of a single mode from the 22 transfer function measurements. These are remarkably good when one considers that the resolution between spectral lines is 10 Hz. Working in a narrower bandwidth or using more data points to describe each transfer function should give better results.

Table 3 contains the corresponding normalized mode shape vectors from the two tests.

CONCLUSIONS

The results indicate that by applying an analytical transfer function expression through least squares estimation to measured data from linearly behaving (small displacements) structures, modal parameters consistent with the theory can be obtained. The vibrations specialist must be continually aware however of the important assumptions necessary for obtaining valid modal results from test specimens.

REFERENCES

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2. Flannelly, W.G. McGarvey, J.H., Berman, A., "A Theory of Identification of the Parameters in the Equations of Motion of a Structure Through Dynamic Testing" paper No. C­1 Symposium on Structural Dynamics, U. of Technology, Loughborough, England, March 1970

3. Thoren, A.R. "Derivation of Mass and Stiffness Matrices form Dynamic Test Data", AIAA paper 72­346, 13th Structures, Structural Dynamics, and Materials Conf., San Antonio, Texas April 12, 1972

4. Hurty, W.C. and Rubinstein, M.F. "Dynamics of Structures" Printice­Hall Inc., Englewood Cliffs, N.J., 1965

5. Caughey, T.K., O'Kelly, M.E.J. "General Theory of Vibration of Damped Linear Dynamic Systems" Cal. Inst. of Tech., Dynamics Lab., Pasadena, California, June 1963