# Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions.

 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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