Please use this identifier to cite or link to this item: https://repository.cihe.edu.hk/jspui/handle/cihe/2955
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dc.contributor.authorLeung, Andrew Yee Taken_US
dc.date.accessioned2022-04-07T09:10:09Z-
dc.date.available2022-04-07T09:10:09Z-
dc.date.issued2001-
dc.identifier.urihttps://repository.cihe.edu.hk/jspui/handle/cihe/2955-
dc.description.abstractThe solution of <i>ż=Az</i> is <i>z(t)</i>=exp(<i>At</i>)z<sub>0</sub>=E<sub>t</sub>z<sub>0</sub>, z<sub>0</sub>=z(0)</i>. Since <i>z</i>(<i>2t</i>)=E<sub>2t</sub>z<sub>0</sub>=E<sup>t</sup><sub>2</sub>z<sub>0</sub></i>, <i>z</i>(<i>4t</i>)=E<sub>4t</sub>z<sub>0</sub>=E<sup>2t</sup><sub>2</sub>z<sub>0</sub></i>, etc., one function evaluation can double the time step. For an <i>n</i>-degree-of-freedoms system, A is a 2<i>n</i> matrix of the <i>n</i>th-order mass, damping and stiffness matrices <i>M</i>, <i>C</i> and <i>K</i>. If the forcing term is given as piecewise combinations of the elementary functions, the force response can be obtained analytically. The mean-square response <i>P</i> to a white noise random force with intensity W(<i>t</i>) is governed by the Lyapunov differential equation: <i>Ṗ=AP+PA<sup>T</sup>+W</i>. The solution of the homogeneous Lyapunov equation is <i>P</i>(<i>t</i>)=exp(<i>At</i>) <i>P</i><sub>0</sub></i> exp(<i>A</i><sup>T</sup><i>t</i>), <i>P</i><sub>0</sub>=<i>P</i>(0). One function evaluation can also double the time step. If <i>W(t)</i> is given as piecewise polynomials, the mean-square response can also be obtained analytically. In fact, exp(<i>At</i>) consists of the impulsive- and step-response functions and requires no special treatment. The method is extended further to coloured noise. In particular, for a linear system initially at rest under white noise excitation, the classical non-stationary response is resulted immediately without integration. The method is further extended to modulated noise excitations. The method gives analytical mean-square response matrices for lightly damped or heavily damped systems without using modal expansion. No integration over the frequency is required for the mean-square response. Four examples are given. The first one shows that the method include the result of Caughy and Stumpf as a particular case. The second one deals with non-white excitation. The third finds the transient stress intensity factor of a gun barrel and the fourth finds the means-square response matrix of a simply supported beam by finite element method.en_US
dc.language.isoenen_US
dc.publisherJohn Wiley & Sonsen_US
dc.relation.ispartofInternational Journal for Numerical Methods in Engineeringen_US
dc.titleFast matrix exponent for deterministic or random excitationsen_US
dc.typejournal articleen_US
dc.identifier.doi10.1002/1097-0207(20010120)50:2<377::AID-NME29>3.0.CO;2-E-
dc.contributor.affiliationSchool of Computing and Information Sciencesen_US
dc.relation.issn1097-0207en_US
dc.description.volume50en_US
dc.description.issue2en_US
dc.description.startpage377en_US
dc.description.endpage394en_US
dc.cihe.affiliatedNo-
item.cerifentitytypePublications-
item.fulltextNo Fulltext-
item.languageiso639-1en-
item.openairecristypehttp://purl.org/coar/resource_type/c_6501-
item.grantfulltextnone-
item.openairetypejournal article-
crisitem.author.deptYam Pak Charitable Foundation School of Computing and Information Sciences-
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