Closed-form solutions of non-uniform axially loaded beams using Lie symmetry analysis
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O R I G I NA L PA P E R
Bidisha Kundu · Ranjan Ganguli
Closed-form solutions of non-uniform axially loaded beams using Lie symmetry analysis
Received: 27 December 2019 / Revised: 16 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, the governing differential equation of a beam with axial force is studied using the Lie symmetry method. Considering the inhomogeneous beam and non-uniform axial load, the governing equation is a fourth-order linear partial differential equation with variable coefficients with no closed-form solution. We search for a favourable coordinate system where the governing equation has a simpler-form or a closedform solution. A favourable coordinate transformation is found using the Lie transformation group method. The system of determining equations for the governing equation of a beam with non-uniform axial load is derived and then solved to find a favourable coordinate system dependent on the spatially variable stiffness, mass, and axial force. The class of non-uniform axially loaded beams which have a closed-form solution is determined. The fixed-free boundary condition is imposed to find the invariant closed-form solution. A comparison between the analytical solution derived by the Lie symmetry method and the numerical solution is presented. Lie symmetry analysis yields hitherto undiscovered closed-form solutions for non-uniform axially loaded beams.
1 Introduction An elastic beam is a fundamental mathematical model which pervades every corner of physics and engineering. [2,12,19]. The beam deflection is dependent not only on the mass of the beam or external applied force but also on the elastic properties of the material and geometry of the beam. For a non-uniform beam, the equation of motion of the beam is a fourth-order linear partial differential equation with variable coefficients. The equation can be derived from the generalized Hooke’s law and force balance or by minimizing the energy of the system. In this paper, we study an elastic beam with spatially variable axial load. Though the equation is linear in nature, due to the presence of variable coefficients, it is very arduous to get the solution analytically. In [15], the closed-form solutions for fundamental natural frequencies of inhomogeneous vibrating beams under axially distributed loading were presented with different boundary conditions. However, the authors assumed polynomial solutions which are valid for the selected nodes. In this paper, an Euler–Bernoulli beam with axial force is investigated. This fourth-order partial differential equation with variable coefficients is studied using the Lie method. The similarity solution of this equation is also found. The model, an elastic beam with axial load, plays a primary role in the dynamical system of the helicopter, the wind turbine, musical instruments, tall buildings, and columns. However, the integrability in a finite number of terms or obtaining an exact solution of the governing equation of this model is an open question. T
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