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Probabilistic Modelling for Frequency Response Functions and Transmissibility Functions with Complex Ratio Statistics

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Probabilistic Modelling for Frequency Response Functions and Transmissibility Functions with Complex Ratio Statistics Probabilistic Modelling for Frequency Response Functions and Transmissibility Functions with Complex Ratio Statistics Meng-Yun Zhao Department of Civil Engineering, Hefei University of Technology, China. E-mail: [email protected] Wang-Ji Yan* Institute for Aerospace Technology & The Composites Group, The University of Nottingham, United Kingdom. Department of Civil Engineering, Hefei University of Technology, China. E-mail: [email protected] Wei-Xin Ren Department of Civil Engineering, Hefei University of Technology, China. E-mail: [email protected] Michael Beer Institute for Risk and Reliability, Leibniz Universität Hannover, Germany. E-mail: [email protected] The distributions of ratios of random variables arise in many applied problems such as in structural dynamics working with frequency response functions (FRFs) and transmissibility functions (TFs). When analysing the distribution properties of ratio random variables through the definition of probability density functions (PDF), the problem is usually accompanied by multiple integrals. In this study, a unified solution is presented to efficiently calculate the PDF of a ratio random variable with its denominator and numerator specified by arbitrary distributions. With the use of probability density transformation principle, a unified expression can be derived for the ratio random variable by reducing the concerned problem into two-dimensional integrals. As a result, the PDFs of the ratio random variable can be efficiently computed by using effective numerical integration techniques. Finally, based on the vibration tests performed on the Alamosa Canyon Bridge, the proposed method is applied to the data to quantify the uncertainty of FRFs and TFs. Keywords : Frequency response function, transmissibility function, probability density function, ratio distribution, numerical integration, structural dynamics 1. Introduction ratio random variables is restricted by assuming that the denominator and numerator are Frequency response functions (FRF) and independent and follow the same distribution. transmissibility functions (TF) are viewed as the More importantly, the available theories are not most fundamental functions in structural applicable to treating complex-valued ratio dynamics (Yan and Ren, 2016a; 2016b; 2018a). random variables. To address this issue, new They have been widely regarded to be powerful theorems have been obtained on multivariate candidates for structural health monitoring (Maia circularly symmetric complex Gaussian ratio et al., 2011; Mao, 2012). Due to the inherent distribution as well as some useful properties randomness of measurements and variability of deduced as corollaries and lemmas are proved environmental conditions, uncertainty impacts mathematically (Yan and Ren, 2016a;2016b). their applications significantly. FRFs and TFs are Unfortunately, the new distribution is limited to complex-valued ratio random variables composed tackling the ratio random variables with of both real and imaginary parts. Therefore, it is denominator and numerator following complex- important to explore new approaches for valued Gaussian distribution with zero mean. quantifying uncertainty of complex ratio random In practical applications it has been proven that variables. the FFT coefficients may deviate from a Gaussian Currently, ratio distributions have been studied distribution, and the circularly symmetric by a large panel of researchers in many fields like complex normal ratio distribution may cause biological and physical sciences and unexpected errors during the process of econometrics (Hinkley, 1969; Korhonen and uncertainty quantification (Yan and Ren, 2018b). Narula, 1989). However, most of the research on Therefore, one needs to propose a more versatile Proceedings of the 29th European Safety and Reliability Conference. Proceedings of the 29th European Safety and Reliability Conference. Edited by Michael Beer and Enrico Zio Copyright c 2019 European Safety and Reliability Association. Published by Research Publishing, Singapore. ISBN: 978-981-11-2724-3; doi:10.3850/978-981-11-2724-3 0827-cd 2714 Proceedings of the 29th European Safety and Reliability Conference 2715 way to compute the ratio distribution following To avoid direct using of the definition to arbitrary probability distributions. calculate multiple CDF and obtaining complex To address the issue, this study aims at ratio distribution function using partial proposing a unified approach which can derivatives, we introduce the principle of efficiently calculate the PDF of complex ratio probabilistic transformation of random vectors. random variables with its denominator and As a result, complex-valued ratio random variable numerator specified by arbitrary probability following arbitrary distribution can reduce to a distributions by reducing the concerned problem problem with a double dimension integral. into two-dimensional integrals. The formula can For the convenience of illustration, the FFT be evaluated by using effective numerical coefficients are denoted by . integration techniques. According to the Assume that FRF and TF at any frequency point் characteristics of the unified integral formula are mathematically denoted as ܇ ൌ ሼܻଵ ǡ, ܻwhich଴ሽ obtained, 2-dimensional Gaussian-Hermite based is defined as the ratio of FFT coefficient of to on sparse-grid quadrature rule is employed to . The frequency point is omittedܷ ൌ ܻ ଵhere.Ȁܻ଴ Direct calculate the solution efficiently. Using the application of principle of probabilisticܻଵ vibration tests performed on the Alamosa Canyon transformationܻ଴ of random vectors is impossible in Bridge, the uncertainties of FRFs and TFs are that the order of and are not equal. To address quantified based on the proposed method. this difficulty, a new random vector with its PDF denoted by܇ ܷ is constructed as ۿ .(Definition of FRF and TF (Yan and Ren, 2016a .2 ۿ ൌ ۿIn such் a way, one݌ ሺcanࢗሻ relate random vectors The FRF denoted by between the ଴ output of the i-th dof ( ) and the input andሼܻ ǡ ܷሽ through a linear transformation . ௜ǡ௝ ௞ ۿሺ is߱ definedሻ as (Yan According to the principle of probabilistic( ܪ ) applied at the j-th dof ܇܅ ൌ ۿ ܇ ௜ ௞ and Ren, 2018): ܺ ሺ߱ ሻ transformation of random vectors (Yan and Ren, ܨ௝ሺ߱௞ሻ 2016a) ܺ௜ሺ߱௞ሻ ܪ௜ǡ௝ሺ߱௞ሻ ൌ ሺͳሻ ݌ࡽሺࢗሻ ൌ ݌ࢅሺ࢟ሻȀȁܬௐȁ ሺ ͵ሻ A TF is definedܨ௝ ሺas߱ ௞theሻ ratio of FFT Here denotes Jacbian coefficient of an arbitrary response to a Ը Ա ௜ǡ௢ ௞ matrix. It ௐis worthܟ notingെܟ that can be referenceܶ responseሺ߱ ሻ , which is a complex- ܬ ൌ ൤ Ա Ը ൨ ௜ ௞ rearranged as ܟ ܟ valued random variable composed ofܺ ሺboth߱ ሻ real ȁܬௐȁ and imaginary partsܺ ௢ሺwhich߱௞ሻ are correlated with each other (Yan and Ren, 2016a,b): ିଶ thereௐ ȁisൌ a functionalȁݕ଴ȁ relationshipሺͶሻ ܬIt is noted thatȁ݀݁ݐ between and . And considering Eq.(3) and (4), can be ௜ ௞ ଴ ଴ ௜ǡ௢ ௞ ܺ ሺ߱ ሻ expressed܇ ൌ ܻ ܃෩ ൌ in ܻ termsሼ૚܃ሽ of random܇ variableܷ and ۿ ሺ߱ ሻ ൌ ሺʹሻ ܶ where all ‘ ’ shown inܺ ௢ሺthe߱௞ ሻbracket denote random vector as ݌ ሺࢗሻ ଴ frequency. ܻ ߱௞ ܷ As is seen from Eq.(1) and (2), it is not difficult ଶ for one to figure out that FRF and TF are ࡽ ଴ ࢅ ଴ As a result,݌ theሺࢗ ሻPDFൌ ȁ ݕof ȁ ݌ is ሺgivenݕ ࢛෥ሻ by ሺ ͷሻ complex-valued ratio random variables composed of both real and imaginary parts which are ܷ correlated with each other. Based on the unique ାஶ ାஶ mathematical structure of the two classical ଶ Ը Ա functions, a unified approach will be proposed in ݌ࢁሺ࢛ሻ ൌ න න ȁݕ଴ȁ ݌ࢅሺݕ଴࢛෥ሻ ݀ݕ ଴ ݀ݕ ଴ ሺ͸ሻ the following sections to compute the PDF of 4. Numericalିஶ Solutionିஶ complex ratio random variables. As is seen from Eq.(6), whether closed-form analytical solution is available or not depends on . PDF of Complex Ratio Random Variable the formula . Obviously, it is highly non- ͵ ෥ሻܝሺݕ଴܇݌ 2716 Proceedings of the 29th European Safety and Reliability Conference trivial to obtain the closed-form analytical well as the TF ( ) observed in detail by solution for arbitrary ratio random variables. One taking the ratio of to are natural way to address the problem mentioned in ܶଶ଴ǡଵሺ߱௞ሻ considered as examples.ଶ଴ ௞ is ଵ the௞ FFT the above is to resort to efficient numerical coefficient of responseܺ atሺ ߱theሻሺ ௞ሻfourthܺ ሺ߱position,ሻ algorithms. It is not difficult to figure out that while , are definedܺସ in the same computing the PDF for a complex-valued ratio way. is the FFT coefficient of input at the random variable involves calculating the 2- third position.ܺଶ଴ ሺ߱௞ሻ Inܺ ଵFig.ሺ߱௞ 2ሻ and Fig. 4, the red solid dimensional Gaussian-Hermite. The Gauss- line andܨଷሺ ߱the௞ሻ green dotted line denotes the t- Hermite quadrature formula for a 2-dimensional location scale and normal distribution functions of integral is given by the real part and imaginary part of and , while the histograms denote the ସ ௞ మ మ probability mass functions from 330 segments.ܺ ሺ߱ ሻ As ାஶ ାஶ ି൫௫భା௫ మ൯ ଶ଴ ௞ ଵ ଶ ଵ ଶ isܺ seenሺ߱ ሻfrom these figures, one can find that t ׬ିஶ ׬ ିஶ ݂ሺݔ ǡ ݔ ሻ ݁ ݀ݔ ݀ݔ ௠భ ௠మ distribution can fit the histograms better than భ మ ௫೔ ௫೔ ௜భ ௜మ As ൎisσ seen௜ ୀଵ σ from௜ ୀଵ ߱Eq.(7),భ ߱ మthe݂൫ݔ totalǡ ݔ number൯ ሺ ͹ofሻ Gaussian distribution. Therefore, the FRF and TF points to be calculated increases exponentially at some frequency points will employ complex t with the dimension. Therefore, the computational ratio probabilistic models. burden also increases significantly with the The theoretical values of the marginal PDFs of dimension. To address the drawbacks, a fast the real part and imaginary part of at quadrature approach based on the sparse-grid rad/s and at rad/s are ସǡଷ ௞ theory will be employed to address the curse of computed using Eq. (6) with the helpܪ of ሺEq.(7),߱ ሻ ଶ଴ǡଵ ௞ dimensionality problem. The locations and whoseͲǤͳ͵ͻߨ results are ܶshownሺ߱ ሻin Fig.͵Ǥͳͺ͵ߨ 3 and 5 by weights of the univariate quadrature points with a comparing to the histograms. It can be seen from range of accuracy levels are determined by Figs. 3 and 5 that the solid curves agree well with asymptotic approximations method.
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