Parameter Estimation from Frequency Response Measurements Using Rational Fraction Polynomials

This is a new formulation which overcomes many of the numerical analysis problems associated with an old least squared error parameter estimation technique. Overcoming these problems has made this technique feasible for implementation on mini-computer based measurement systems. This technique is not only useful in modal analysis applications for identifying the modal parameters of structures, but it can also be used for identifying poles, zeros and resonances of combined electro-mechanical servo-systems. INTRODUCTION During the last ten years, a variety of FFT-based two channel digital spectrum analyzers have become commercially available which can make frequency response measurements. These analyzers are being used to make measurements on mechanical structures, in order to identify their mechanical resonances, or modes of vibration. Likewise, they are being used to measure the dynamic characteristics of electronic networks, and of combined electro-mechanical servo-systems. One of the key advantages of the digital analyzers is that the measurements which they make, such as frequency response functions, are in digital form, i.e. computer words in a digital memory. Therefore, these measurements can be further processed to identify the dynamic properties of structures and systems. The frequency response function (FRF) is, in general, a complex valued function or waveform defined over a frequency range. (See Figure 4.) Therefore, the process of identifying parameters from this type of measurement is commonly called curve fitting, or parameter estimation. This paper presents the results of an algorithm development effort which was begun back in 1976. At that time, we were looking for a better method for doing curve fitting in a mini-computer based modal analysis system. This type of system is used to make a series of FRF measurements on a structure, and then perform curve fitting on these measurements to identify the damped natural frequencies, damping, and mode shapes of the predominant modes of vibration of the structure. The three main requirements for a good curve fitting algorithm in a measurement system are 1) execution speed, 2) numerical stability, and 3) ease of use. Previous to this development effort, we had experimented with a well known curve fitting algorithm called the Complex Exponential, or Prony algorithm. This algorithm has undergone a lot of refinement ([2], [3]) and is computationally very efficient and numerically stable in 16bit machines. However, it curve fits the impulse response function instead of the FRF. The impulse response can be obtained by taking the Inverse Fourier transform of the FRF. When the FFT is used to obtain the impulse response from an FRF measurement, a potentially serious error can occur, which is called wrap around error, or time domain leakage. This error is caused by the truncated form (i.e. limited frequency range) of the FRF measurement, and distorts the impulse response as shown in Figure 1. Hence, we sought to develop an algorithm with some of the same characteristics as the complex exponential method, (e.g., it is easy to use along with being numerically stable), FIGURE 1. Impulse Response with Leakage. Presented at 1 IMAC Conference, Orlando, FL November, 1982 page 2 but that curve fits the FRF measurement data directly in the frequency domain. If it is assumed that the frequency response measurement is taken from a linear, second order dynamical system, then the measurement can be represented as a ratio of two polynomials, as shown in Figure 6. In the process of curve fitting this analytical form to the measurement data, the unknown coefficients of both the numerator and denominator polynomials, (ak, k=0,...,m) and (bk, k=0,...,n), are determined. It is shown later that this curve fitting can be done in a least squared error sense by solving a set of linear equations, for the coefficients. The greatest difficulty with curve fitting polynomials in rational fraction form is that the solution equations are illconditioned and hence are difficult, if not impossible to solve on a mini-computer, even for simple cases. To curve fit with an m order numerator and n order denominator polynomial, (m+n+1) simultaneous equations must be solved. This is equivalent to inverting an (m+n+1) matrix. Part of the problem stems from the dynamic range of the polynomial terms themselves. For instance, the highest order term in a 12 order polynomial evaluated at a frequency of 1 kHz, is on the order of 10 to the 36 power, which borders on the standard numerical capability of many 16 bit mini-computers. We will see later that this problem can be minimized by re scaling the frequency values. However, this single step doesn't change the ill-conditioned nature of the solution equations. We did find, though, that the use of orthogonal polynomials removes much of the ill-conditioning, and at the same time reduces the number equations to be solved to about half the number of equations of the ordinary polynomial case. Much of the discussion in this paper, then, centers on the reformulation of the solution equations in terms of orthogonal polynomials, and generation of the polynomials themselves. An alternative formulation, which yields an estimate of the characteristic polynomial from multiple measurements is also included. Finally, some examples of the use of the curve fitter are given. Some of the problems which are common to all curve fitters, such as measurement noise, frequency resolution, and the effects of resonances which lie outside of the analysis band are discussed. companion paper (Reference [4]) discusses in more detail, these and other problems which can occur when curve fitting FRF measurements. MODELING SYSTEM DYNAMICS IN THE LAPLACE DOMAIN The dynamics of a mechanical structure can be modeled with a Laplace domain modal, as shown in Figure 2. In this model, the inputs and responses of the structure are represented by their Laplace transforms, Time domain derivatives (i.e. velocity and acceleration) do not appear explicitly in the Laplace domain model but are accounted for in the transfer functions of the structure. These transfer functions are contained in a transfer matrix and contain all of the information necessary to describe structural responses as functions of externally applied forces. Using a Laplace domain model, a structure is excited by several different input forces, then its transformed response is a summation of terms, each term containing one of the transformed input forces multiplied by the transfer function between the input point (degree-of-freedom) and the desired response point. Transfer Function of a Single Degree-of-Freedom System The Laplace variable is a complex variable, normally denoted by the letter s. Since the transfer function is a function of the s-variable, it too is complex valued, (i.e. for every complex value of s, the transfer function value is a complex number). Plots of a typical transfer function on the s-plane are shown in Figure 3. Because it is complex, Presented at 1 IMAC Conference, Orlando, FL November, 1982 page 3 the transfer function can be represented by its real and imaginary parts, or equivalently by its magnitude and phase. Note that the magnitude of the transfer function goes to infinity at two points in the s-plane. These singularities are called the Poles of the transfer function. These poles define resonant conditions on the structure which will "amplify" an input force. The location of these poles in the s-plane is defined by a Frequency and Damping value as shown in Figure 5. Hence the σ and jω axes of the s-plane have become known as the damping axis and the frequency axis respectively. Note also in Figure 5 that the transfer function is only plotted over half of the s-plane, i.e. it is not plotted for any positive values of damping. This was done to give a clear picture of the transfer function values along the frequency axis. The Frequency Response Function In a test situation we do not actually measure the transfer function over the entire s-plane, but rather its values along the frequency axis. These values are known as the frequency response function, as shown in Figure 3. The analyzers compute the FRF by computing the Fourier transform of both the input and response signals, and then forming the ratio of response to input in the frequency domain. The resulting function is the same as evaluating the system's transfer function for s=jω. Since the transfer function is an "analytic" function, its values throughout the s-plane can be inferred from its values along the frequency axis. A dynamic frequency domain model, similar to the Laplace domain model, can also be built using frequency response functions. The form of the frequency domain model is exactly the same as the Laplace domain model, but with frequency response functions replacing transfer functions and Fourier transforms replacing Laplace transforms of the structural inputs and responses. The frequency response function, being complex valued, is represented by two numbers at each frequency. Figure 4 shows some of the alternative forms in which this function is commonly plotted. The so called CoQuad plot, or real and imaginary parts vs. frequency, derives its origin from the early days of swept sine testing when the real part was referred to as the coincident waveform (that portion of the response that is in phase with the input) and the imaginary part as the quadrature waveform (that portion of the response that is 90 degrees out-of-phase with the input). The Bode plot, or log magnitude and phase vs. frequency, is named after H.W. Bode who made many contributions to the analysis of frequency response functions. (Many of Bode's techniques involved plotting these functions along a log frequency axis.) Presented at 1 IMAC Conference, Orlando, FL November, 1982 page 4 The Nyquist plot, or real vs. Imaginary part, is named after the gentleman who popularized its use for determining the stability of linear systems. The Nichols plot or log magnitude vs. phase angle is named after N.B. Nichols who used such plots to analyze servo-mechanisms. It is important to realize that all of these different forms of the frequency response function contain exactly the same information, but are presented in different forms to emphasize certain features of the data. ANALYTICAL FORMS OF THE FREQUENCY RESPONSE The FRF can he represented either in rational fraction or partial fraction form, as shown in Figure 6. Rational Fraction Form The rational fraction form is merely the ratio of two polynomials, where in general the orders of the numerator and denominator polynomials are independent of one another. The denominator polynomial is also referred to as the characteristic polynomial of the system. Recalling that the FRF is really the transfer function evaluated along the frequency axis, the poles of the transfer function correspond to values of the s-variable for which the characteristic polynomial is zero, i.e. the transfer function is infinite. These values of s are also called the roots of the characteristic polynomial. Similarly, the roots of the numerator polynomial are the values of the s-variable where the transfer function is zero and are therefore called the zeros of the transfer function. Hence, by curve fitting the analytical form in equation (1) to FRF data, and then solving for the roots of both the numerator and characteristic polynomials, the poles and zeros of the transfer function can be determined. Poles and zeros are typically used to characterize the dynamics of electronic networks and servo-systems, Partial Fraction Form For resonant systems, that is, systems where the poles are not located along the damping axis, the FRF can also be represented in partial fraction form. This form clearly shows the FRF in terms of the parameters which describe its pole locations. For a model with n-degrees-of-freedom, it is clear that the FRF contains n-pole pairs. In this form, the numerator simply becomes a pair of constants, called residues, which also occur as complex conjugate pairs. Every pole has a different residue associated with it. In modal analysis, the unknown parameters of the partial fraction form, i.e. the poles and residues, are used to characterize the dynamics of structures, PROBLEM FORMULATION The curve fitting problem consists of finding the unknown (ak, k=0,...,m) and (bk, k=0,...,n) such that the error between the analytical expression (1) in Figure 6 and an FRF measurement is minimized over a chosen frequency range. To begin the problem formulation, we need to define an error criterion. ANALYTICAL FORMS OF THE FREQUENCY RESPONSE FUNCTION Rational Fraction Form