Numerically Stable form of Green’s Function for a Free-Free Uniform Timoshenko Beam
Beam models are widely applied in civil engineering, transport, and industry because the beams are basic structural elements. When dealing with the high-order modes of beam in the context of applying the modal analysis method, the numerical instability issue affects the numeric simulation accuracy i...
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| Format: | Article |
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MDPI AG
2022-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/1/86 |
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| author | Traian Mazilu |
| author_facet | Traian Mazilu |
| author_sort | Traian Mazilu |
| collection | DOAJ |
| description | Beam models are widely applied in civil engineering, transport, and industry because the beams are basic structural elements. When dealing with the high-order modes of beam in the context of applying the modal analysis method, the numerical instability issue affects the numeric simulation accuracy in many boundary conditions. There are two solutions in literature to overcome this shortcoming, namely refinement of the asymptotic form for the high order modes and reshaping the terms within the equation of the modes to eliminate the source of the numerical instability. In this paper, the numerical instability issue is signalled when the standard form of Green’s function, which includes hyperbolic functions, is applied to a free-free Timoshenko length-long beam. A new way is proposed based on new set of eigenfunctions, including an exponential function, to construct a new form of Green’s function. To this end, it starts from a new general form of Green’s function and the characteristic equation is obtained; then, based on the boundary condition, the Green’s function associated to the differential operator of the free-free Timoshenko beam is distilled. The numerical stability of the new form of the Green’s function is verified in a numerical application and the results are compared with those obtained by using the standard form of the Green’s function. |
| first_indexed | 2024-03-09T12:08:29Z |
| format | Article |
| id | doaj.art-3c1ce2f3910349a5b9dd25498909e427 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T12:08:29Z |
| publishDate | 2022-12-01 |
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| spelling | doaj.art-3c1ce2f3910349a5b9dd25498909e4272023-11-30T22:54:54ZengMDPI AGMathematics2227-73902022-12-011118610.3390/math11010086Numerically Stable form of Green’s Function for a Free-Free Uniform Timoshenko BeamTraian Mazilu0Department of Railway Vehicles, University Politehnica of Bucharest, Splaiul Independenței, 313, 060032 Bucharest, RomaniaBeam models are widely applied in civil engineering, transport, and industry because the beams are basic structural elements. When dealing with the high-order modes of beam in the context of applying the modal analysis method, the numerical instability issue affects the numeric simulation accuracy in many boundary conditions. There are two solutions in literature to overcome this shortcoming, namely refinement of the asymptotic form for the high order modes and reshaping the terms within the equation of the modes to eliminate the source of the numerical instability. In this paper, the numerical instability issue is signalled when the standard form of Green’s function, which includes hyperbolic functions, is applied to a free-free Timoshenko length-long beam. A new way is proposed based on new set of eigenfunctions, including an exponential function, to construct a new form of Green’s function. To this end, it starts from a new general form of Green’s function and the characteristic equation is obtained; then, based on the boundary condition, the Green’s function associated to the differential operator of the free-free Timoshenko beam is distilled. The numerical stability of the new form of the Green’s function is verified in a numerical application and the results are compared with those obtained by using the standard form of the Green’s function.https://www.mdpi.com/2227-7390/11/1/86Green’s functionhyperbolic functionexponential functiondifferential operatornumerical instabilityfree-free Timoshenko beam |
| spellingShingle | Traian Mazilu Numerically Stable form of Green’s Function for a Free-Free Uniform Timoshenko Beam Mathematics Green’s function hyperbolic function exponential function differential operator numerical instability free-free Timoshenko beam |
| title | Numerically Stable form of Green’s Function for a Free-Free Uniform Timoshenko Beam |
| title_full | Numerically Stable form of Green’s Function for a Free-Free Uniform Timoshenko Beam |
| title_fullStr | Numerically Stable form of Green’s Function for a Free-Free Uniform Timoshenko Beam |
| title_full_unstemmed | Numerically Stable form of Green’s Function for a Free-Free Uniform Timoshenko Beam |
| title_short | Numerically Stable form of Green’s Function for a Free-Free Uniform Timoshenko Beam |
| title_sort | numerically stable form of green s function for a free free uniform timoshenko beam |
| topic | Green’s function hyperbolic function exponential function differential operator numerical instability free-free Timoshenko beam |
| url | https://www.mdpi.com/2227-7390/11/1/86 |
| work_keys_str_mv | AT traianmazilu numericallystableformofgreensfunctionforafreefreeuniformtimoshenkobeam |