Best conditioned common denominator transfer function matrix estimation in the frequency domain

Multi-Input-Multi-Output vibration testing typically requires the determination of inputs to achieve desired response at multiple locations. First, the responses due to each input are quantified in terms of complex transfer functions in the frequency domain. In this study, two Inputs and five Responses were used leading to a 5 × 2 transfer function matrix. Inputs corresponding to the desired Responses are then computed by inversion of the rectangular matrix using Pseudo-Inverse techniques that involve least-squared solutions. It is important to understand and quantify the various sources of errors in this process toward improved implementation of Multi-Input-Multi-Output testing. In this article, tests on a cantilever beam with two actuators (input controlled smart shakers) were used as Inputs while acceleration Responses were measured at five locations including the two input locations. Variation among tests was quantified including its impact on transfer functions across the relevant frequency domain. Accuracy of linear superposition of the influence of two actuators was quantified to investigate the influence of relative phase information. Finally, the accuracy of the Multi-Input-Multi-Output inversion process was investigated while varying the number of Responses from 2 (square transfer function matrix) to 5 (full-rectangular transfer function matrix). Results were examined in the context of the resonances and anti-resonances of the system as well as the ability of the actuators to provide actuation energy across the domain. Improved understanding of the sources of uncertainty from this study can be used for more complex Multi-Input-Multi-Output experiments.

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A Numerical Evaluation of the Quadratic Transfer Function for a Floating Structure

Volume 7A: Ocean Engineering ◽

10.1115/omae2019-95620 ◽

2019 ◽

Author(s):

Zhitian Xie ◽

Yujie Liu ◽

Jeffrey Falzarano

Keyword(s):

Transfer Function ◽

Frequency Domain ◽

Transfer Functions ◽

Second Order ◽

Wave Frequency ◽

Wave Forces ◽

Floating Structure ◽

Transfer Function Matrix ◽

Wave Induced ◽

Function Matrix

Abstract The second order force of a floating structure can be expressed in terms of a time independent quadratic transfer functions along with the incident wave elevation, through which it is possible to evaluate the second order wave exciting forces in the frequency domain. Newman’s approximation has been widely applied in approximating the elements of the quadratic transfer function matrix while numerically evaluating the second order wave induced force. Through Newman’s approximation, the off-diagonal elements can be numerically approximated with the diagonal elements and thus the numerical calculation efficiency can be enhanced. Newman’s approximation assumes that the off-diagonal elements do not change significantly with the wave frequency and that hydrodynamic phenomenon regarding the low difference frequency are usually of interest. However, it is obviously less satisfying when an element that is close to the diagonal line in the quadratic transfer function matrix shows an extremum if the corresponding wave frequency is close to the natural frequency of the certain motion. In this paper, the full derivation and expression of the second order wave forces and moments applied to a floating structure have been presented, through which the numerical results of the quadratic transfer function matrix including the diagonal and the off-diagonal elements will be illustrated. This work will present the basis of numerically evaluating the second order forces in the frequency domain. The comparisons among various approximations regarding the second order forces in deep water will also be presented as a meaningful reference.

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