Global Curve Fitting of Frequency Response Measurements using the Rational Fraction Polynomial Method

The latest generation of FFT Analyzers contain still more and better features for excitation, measurement and recording of frequency response functions (FRF's) from mechanical structures. As measurement quality continues to improve, a larger variety of curve fitting methods are being developed to handle a set of FRF measurements in a global fashion. These approaches can potentially yield more consistent modal parameter values than curve fitting individual measurements independently. In this paper, a new formulation of the Rational Fraction Polynomial method is given which can globally curve fit a set of FRF measurements. The pros and cons of this approach are discussed, and an example is included to compare the results of this method with a local curve fitting method. INTRODUCTION Physically speaking, a mode of vibration of a structure is characterized by a so called "natural" or "resonant' frequency at which the structure's predominant motion is a well defined waveform, called the "mode shape". A mode is the manifestation of energy which is trapped within the boundaries of the structure, and cannot readily escape. When a structure is excited, its linear response can be shown to be a function of the combined motions of its modes of vibration. That is, the overall motion can be represented as a linear combination of the motions of each of the modes. Likewise, when the excitation source is removed from the structure, the trapped energy within it will slowly decay out until it no longer vibrates. The rate at which energy decays out of the structure is controlled by the amount of damping in the structure. Damping is also a modal property, each mode having a certain value of damping associated with it. That is, the motion comprised of heavily damped modes will decay out more quickly than that part of the motion comprised of more lightly damped modes. Modes of vibration can be observed in practically any vibrating structure. When we measure the vibration of a structure and decompose the vibration signal into its frequency spectrum, the modes of vibration are evidenced by peaks in the spectrum. (Other peaks may be present in the spectrum due to large cyclical excitation forces). The modal peaks, however, will appear in practically any measurement made from any point on the structure. In summary, then, each mode is a global property of the structure. Each mode is defined by a natural (or modal) frequency, a value of (modal) damping, and a mode shape. Since modes are properties of the structure, itself, and are independent of the type of excitation force used to excite it, they should be identified from measurements which are also independent of the type of excitation. The Frequency Response Function (FRF) is such a measurement for linear systems. The FRF is essentially a “normalized” measure of structural response. That is, it is the ratio of a response spectrum divided by the spectrum of the excitation which causes the response. Hence, the FRF is a measure of the dynamic properties between two degrees-of-freedom (DOF's) of a structure; the excitation point (and direction) and the response point (and direction). Again, the modes of the structure are indicated by the peaks in the measurement, with at least one mode defined by each peak. Figure 1 shows in simplified form how the first three modes of a beam are identified from a set of FRF measurements made from the beam. The figure shows the imaginary part of each of the FRF measurements which were made between some (arbitrary) excitation point, and each of the Presented at 3 IMAC Conference, Orlando, FL January, 1985 page 2 response points marked with X's. In this case, responses were measured only in the vertical direction and with a transducer that measured either displacement or acceleration. Alternatively, the FRF's could have been measured by mounting a single response transducer in one (arbitrary) location and exciting the beam at each X, in the vertical direction. The figure shows a modal peak at the same frequency in each measurement, indicating the global nature of modal frequency. The “width” of the modal peak for each mode should also be the same in each measurement, again indicating the global nature of modal damping. Lastly, the mode shape which is defined by assembling the modal peak values from all the measurements, is global in the sense that it is defined for the entire expanse of the structure. Mathematically speaking, modes of vibration are defined by certain parameters of a linear dynamic model for a structure. The dynamic properties of a structure can be written either as a set of differential equations in the time domain, or as a set of equations containing transfer functions in the Laplace (frequency) domain. These equivalent models are shown in Figures 2 and 3. Regardless of which model is used, it can be shown that either model can be written in terms of the same parameters (frequencies, damping, and mode shapes) that describe the modes of vibration. CURVE FITTING FRF's Curve fitting, or Parameter Estimation, is a numerical process that is typically used to represent a set of experimentally measured data points by some assumed analytical function. The results of this curve fitting process are the coefficients, or parameters, that are used in defining the analytical function. With regard to the Frequency Response Function, the parameters that are calculated are its so-called modal parameters (i.e. modal frequency, damping, and residue). The curve fitting process can also be thought of as a data compression process since a large number of experimental values (the FRF measurements) can be represented by a much smaller number of modal parameters. Various forms of the transfer function dynamic model are used to curve fit FRF measurements. The transfer function model is, in effect, evaluated along the frequency axis (i.e. s=jω) during the curve fitting process. The entire transfer function model is shown in Figure 3 , and it is well known [2] from examination of this model that curve fitting of one row or one column of FRF's is sufficient to identify the modal properties of the structure. However, selection of the correct row or column may be very important, depending on the modes of interest, the geometry of the structure, etc. Presented at 3 IMAC Conference, Orlando, FL January, 1985 page 3 Nevertheless, once a set of FRF measurements has been made on a structure, whether they comprise one row or column, or several, the most commonly used method of curve fitting FRF's is to fit them one at a time using one of the analytical forms of the FRF shown in Figure 4. The most commonly used form is the Partial Fraction Form. Most SDOF (or single mode) methods, and various iterative MOOF (multiple mode) methods, are based on this form of the model. RATIONAL FRACTION POLYNOMIAL (RFP) METHOD In a previous paper [1], a curve fitting method based on the Rational Fraction Polynomial form of the FRF was introduced. This MDOF method fits the analytical expression (1) to an FRF measurement in a least-squared error sense, and in the process, the coefficients of the numerator and denominator polynomials are identified. Analytical Forms of the Frequency Response Function Rational Fraction Form