Measurement and modal analysis of transfer functions
By measuring the transfer function, we can observe the amplitude at the measurement point and its phase relative to the excitation point. While some objects exhibit strong linearity, such as metal rods or plates suspended in a free position, where the first and nth measurement data are almost identical, most objects generally involve nonlinear elements such as friction and play. In such cases, averaging is performed to reduce measurement variability and errors. This is similar to the arithmetic mean of a power spectrum, where the error value gradually converges to a constant level with each averaging. At this point, by examining the coherence function, we can determine how much the excitation energy contributes to the measurement point.
The presence of noise in the measurement system will reduce the coherence value. Similarly, the presence of nonlinear elements in the object being measured will also reduce the coherence value. Personally, as a guideline, I check the measurement method and other factors to ensure that the coherence value is 0.9 or higher. In reality, we consider the structural conditions and the support conditions and excitation signals to ensure that the coherence value is maintained in the desired frequency band.
Now, the measured transfer function often has several peaks and valleys.
This means that at the measurement point, there are frequencies at which vibration is more likely to occur and frequencies at which it is less likely to occur.
Furthermore, even at the same frequency (a specific frequency), the magnitude changes when the measurement position is changed, meaning that both the amplitude and phase change simultaneously. By using the magnitude and phase of the specific frequency at each measurement point, we can observe the shape of the vibration mode at that specific frequency. The transfer function measured at each point is merely a sequence of frequency component data, and does not represent the overall characteristics (system characteristics) in mathematical terms; it is simply a sequence of dispersed (discrete) data. However, this data sequence represents the mass M, spring constant K, and damping constant C at each frequency. Therefore, by using curve fitting with amplitude and phase as variables, this transfer function can be expressed as a polynomial with an Nth-order denominator and an M-order numerator.
A simple curve fitting procedure involves specifying the area before and after a particular peak as the calculation range, and determining the number of peaks, thereby determining the degree of the M- and N-degree polynomials in the numerator and denominator.
Starting from the initial equation, we repeatedly perform iterative calculations to find the polynomial constants that approximate the transfer function. From the obtained equation, we can derive the characteristic equation. The characteristic equation is expressed as a characteristic matrix by creating a model (M, K, C) using the eigenvalues (resonance frequency) and eigenvectors (mode shape) of the free vibration, as explained previously. Using this characteristic matrix, we can perform simulations of vibration modes.
In other words, we have been able to convert it into a physical model, and this characteristic matrix (characteristic equation) corresponds to M, K, and C.
Modal analysis involves creating a motion model of the object being measured using M, K, and C, and then performing eigenvalue analysis to generate the resonant frequency and eigenvectors, which then give rise to the mode shape. Measuring the transfer function is the reverse of this process; it involves inversely calculating the characteristic equation and ultimately the equation of motion from the natural frequencies and mode shapes of the transfer function.
Regarding excess mass and excess stiffness
In the case of a continuous body such as a round rod, there is an infinite number of masses, but in calculations of models to determine their properties, the number of mass points is limited to a finite N. Also, when performing curve fitting with a transfer function, the bandwidth is limited by setting upper and lower frequency limits, but how is the low and high frequency data that is cut off in the calculation evaluated? Basically, there are an infinite number of vibration modes. To compensate for these truncated modes, excess mass (effect of omitting low frequencies) and excess stiffness (effect of omitting high frequencies) are used to correct the error in the fitted bandwidth.
(Excerpt from the email newsletter issued on November 18, 2004)
