Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions.

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dc.contributor.author Naz, R.
dc.contributor.author Mahomed, F.M.
dc.date.accessioned 2016-06-06T14:40:07Z
dc.date.available 2016-06-06T14:40:07Z
dc.date.issued 2015
dc.identifier.citation Naz, R. and Mahomed, F.M. 2015. Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions. Mathematical Problems in Engineering. en_ZA
dc.identifier.issn 1024-123X (Print)
dc.identifier.issn 1563-5147 (Online)
dc.identifier.uri http://hdl.handle.net/10539/20433
dc.description.abstract We 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.language.iso en en_ZA
dc.publisher Hindawi Publishing Corporation en_ZA
dc.subject Algebra en_ZA
dc.subject Boundary value problems en_ZA
dc.subject Continuum mechanics en_ZA
dc.subject Differential equations en_ZA
dc.subject Initial value problems en_ZA
dc.subject Lie groups en_ZA
dc.subject Nonlinear equations en_ZA
dc.subject Ordinary differential equations en_ZA
dc.subject Partial differential equations en_ZA
dc.subject Equivalence transformations en_ZA
dc.subject Euler-Bernoulli beam equation en_ZA
dc.subject Fourth order ordinary differential equation en_ZA
dc.subject Infinite dimensional Lie algebras en_ZA
dc.subject Initial and boundary conditions en_ZA
dc.subject Initial-boundary value problems en_ZA
dc.subject Similarity transformation en_ZA
dc.title Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions. en_ZA
dc.type Article en_ZA

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