Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions.

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dc.contributor.authorNaz, R.
dc.contributor.authorMahomed, F.M.
dc.date.accessioned2016-06-06T14:40:07Z
dc.date.available2016-06-06T14:40:07Z
dc.date.issued2015
dc.description.abstractWe study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass density g(x), and the applied load denoted by f(u), a function of transverse displacement u(t,x). The complete Lie group classification is obtained for different forms of the variable lineal mass density g(x) and applied load f(u). The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms of g(x). For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature when g(x) is constant with variable applied load f(u). For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions.en_ZA
dc.identifier.citationNaz, R. and Mahomed, F.M. 2015. Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions. Mathematical Problems in Engineering.en_ZA
dc.identifier.issn1024-123X (Print)
dc.identifier.issn1563-5147 (Online)
dc.identifier.urihttp://hdl.handle.net/10539/20433
dc.language.isoenen_ZA
dc.publisherHindawi Publishing Corporationen_ZA
dc.subjectAlgebraen_ZA
dc.subjectBoundary value problemsen_ZA
dc.subjectContinuum mechanicsen_ZA
dc.subjectDifferential equationsen_ZA
dc.subjectInitial value problemsen_ZA
dc.subjectLie groupsen_ZA
dc.subjectNonlinear equationsen_ZA
dc.subjectOrdinary differential equationsen_ZA
dc.subjectPartial differential equationsen_ZA
dc.subjectEquivalence transformationsen_ZA
dc.subjectEuler-Bernoulli beam equationen_ZA
dc.subjectFourth order ordinary differential equationen_ZA
dc.subjectInfinite dimensional Lie algebrasen_ZA
dc.subjectInitial and boundary conditionsen_ZA
dc.subjectInitial-boundary value problemsen_ZA
dc.subjectSimilarity transformationen_ZA
dc.titleDynamic Euler-Bernoulli Beam Equation: Classification and Reductions.en_ZA
dc.typeArticleen_ZA
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